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Thinker…. Lew is playing darts on a star-shaped dartboard in which two equilateral triangles trisect the sides of each other as shown. Assuming that a dart hits the board, what is the probability that it will land inside the hexagon?

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**Solution to the “Thinker”**

1/2. As shown, each triangle can be reflected to the interior of the hexagon in such a way that the triangle areas are equal to the area of the hexagon. In this manner, the area of the hexagon is half that of the entire dartboard.

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**Functions Domain & Range Increasing & Decreasing**

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**A set of ordered pairs. (x, y)**

What is a relation? A set of ordered pairs. (x, y) Could you represent a relation another way?

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Domain vs. Range Domain- the set of x-coordinates of a relation or function. Range- the set of y-coordinates of a relation or function. Notice anything about the order of the domain/range?

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Functions Discrete- a graph which consists of points which are not connected.

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**A function that is traceable!**

Continuous Data What do you think of when you hear the word continuous? A function that is traceable! Examples: Lines…Parabolas…any others?

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Discontinuous Data What do you think of when you hear the word discontinuous? A function that is not traceable. (must pick up your pencil) Would discrete data be continuous or discontinuous?

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Piecewise Functions -a graph which consists of line segments or pieces of other nonlinear graphs. Would piecewise functions be continuous or discontinuous? Would piecewise functions be discrete?

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**What is a function? A function is a relation in which**

each element of the domain is paired with exactly one element from the range (no duplicate x-values). A function is denoted as f(x) or pronounced “f of x”.

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**Some relations are functions…some are not…**

But…how do we know? Let’s find out!

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**Draw these two graphs. Vertical Line Test!**

Touches it once, it IS a function! Touches >1, it is NOT a function!

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**What if it’s not a graph?? Do the x-values repeat?**

State weather the following relation is a function or not. Do the x-values repeat? No they don’t….YES it is a function! Yes they repeat…NO it’s not a function! That’s too hard to remember, can I just graph the points and use the VLT?

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Example 1 Determine whether the relation {(-1, ), (-1, ), (0, 1)} is a function. Justify your decision in a complete sentence. This relation is not a function since two values of -1 will not pass the vertical line test.

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**Examples State whether each relation or graph below is a function:**

{(1,2),(2,4),(3,5)(0,5)} {(0,4),(2,4),(1,3),(2,5)} Yes No

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Examples State whether each relation or graph below is a function: no yes

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Examples State whether each relation or graph below is a function: (5,5) is open yes no

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**(a, b) a< x <b Open Interval Parenthesis-NOT inclusive**

An open interval is the set of all real numbers that lie strictly between two fixed numbers a and b. Parenthesis-NOT inclusive is always open (a, b) a< x <b ***think about open dots on a number line!

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**[a, b] a< x <b Closed Interval Brackets mean-inclusive**

A closed interval is the set of all real numbers that lie in between and contain both endpoints a and b. [a, b] a< x <b Brackets mean-inclusive **think about closed dots on a number line!

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**(a, b] [a, b) a< x <b a< x <b Half-Open Intervals**

Half-open intervals are intervals that contain one but not both endpoints a and b. (a, b] a< x <b [a, b) a< x <b

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What do we use this for? **Intervals will be used to define the domain and range of given functions or graphs which are continuous and/or increasing and decreasing intervals. OTHER SYMBOLS TO KNOW: U : union symbol used to join more than one interval together. 0 (zero): neither positive nor negation.

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**Number Line Examples -1< x <4 x< 1 or x > 5**

Fill in the missing parts in the chart below. Inequality Interval Notation Graph -1< x <4 x< 1 or x > 5

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Graph Example 2 Determine the domain, range, and continuity of the graph below.

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Graph Example 3 Determine the domain, range, and continuity of the graph below.

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Graph Example 4 Determine the domain, range, and continuity of the graph below.

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Graph Example 5 Determine the domain, range, and continuity of the graph below.

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**Increasing/Decreasing**

Increasing intervals occur when. . .reading a graph left to right, the interval in which the function is rising. Decreasing intervals occur when. . .reading a graph left to right, the interval in which the function is falling.

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**Increasing/Decreasing Example 6**

a. Determine the interval(s) of x in which f(x) in increasing. (between what two x values is the function increasing?)

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**Increasing/Decreasing Example 6**

b. Determine the interval(s) of x in which f(x) in decreasing. (between what two x values is the function decreasing?)

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**Increasing/Decreasing Example 6**

c. Determine the interval(s) of x in which f(x) is positive. (between what two x values are the y-values positive?)

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Reflection Write a question in your notebook about something Mrs. Gromesh taught today that you aren’t 100% on understanding yet.

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